Influence of the network properties on the phase transition of the SIR epidemic model

Out of equilibrium simulations have been used as a technique for characterization of many systems in statistical mechanics. Epidemic models as SI, SIR, SIS and SIRS and have been studied using differential equations and also Monte Carlo simulations. In the latter, usually, a square lattice is used as a network and each vertex is one individual. The positions of the individuals and the interaction between them do not change with time. The phase transitions and the critical exponent theta have been obtained with this kind of network. In this work, we explore the influence of the network allowing it to evolve during the time evolution. This is accomplished by considering each individual as a random walker moving by a fixed quantity in an arbitrary direction. Instantaneous and different networks are thus formed in each iteration. Now the interaction will be defined by the distances between the individuals, making these interactions changes in each iteration. Within this model, we identified the new phase transition, the critical exponent theta and the features of the aggregated weighted network (the weight distribution and correlation of the interactions).